# Cryptography 101

## What is cryptography

• The art of writing or solving codes.
• Comes from the greek kryptos-graphein meaning hidden writing
• The only mathematically sound security measure

## Brief history

### Cesar Cipher

• Used by Cesar for personal communication in the 1st Century BCE
• Simple substitution cipher
• Can be easily cracked based on character frequency

### Enigma

• Used by Nazi Germany during WWII
• Complex substitution cipher
• Fatal flaw that no letter could be itself
• Cracked by Alan Turing and his Team from the UK

### The big change

• Up until this point all ciphers where symmetric
• Means the same key is used to encrypt and decrypt
• How to share the key securely?

### RSA

• The first asymmetric cipher
• Developed in 1977
• Has two keys instead of one

## How does this work

• Relies on one way math functions
• Some mathematical functions are east to do one way, but hard to reverse
• These are:
• Multiplying primes
• Modulus
• Elliptic Curves

### Primes

• Multiplying two primes

\begin{equation} 7 \times 13 = 91 \text{(Easy)} \end{equation}

• Factoring a product of two primes

\begin{equation} a \times b = 68 \text{(Hard)} \end{equation}

### Modulus

• Modulus is the remainder

• Like a clock

\begin{equation} 13 \mod 12 = 1 \end{equation}

\begin{equation} 16 \mod 12 = 4 \end{equation}

• Doing reverse is hard

\begin{equation} a \mod 10 = 5 \end{equation}

• what is a?

• 5, 15, 25, 55, 105, or 900000005?

### Elliptic Curves

• Way out of the scope of this talk
• Basically using curve functions to calculate data
• Think parabolas and such

## RSA

• Relies on Primes and Modulus
• Private and public keys
• Keys are just prime numbers
• You can encrypt with public or private
• Whatever key you do an operation with, you will need the opposite key to reverse it
• If you encrypt with private, you can only decrypt with public

### Example

1. Choose two distinct prime numbers

\begin{equation} p=61 \
q=51 \end{equation}

2. Compute n = pq

\begin{equation} n = 61 \times 53 = 3233 \end{equation}

3. Compute the Carmichael’s totient function of the product as λ(n) = lcm(p − 1, q − 1)

\begin{equation} \lambda(n) = \text{lcm}(60, 52) = 780 \end{equation}

4. Choose any number 1 < e < 780 that is coprime to 780. Choosing a prime number for e leaves us only to check that e is not a divisor of 780.

\begin{equation} e = 17 \end{equation}

5. Compute d, the modular multiplicative inverse of e (mod λ(n))

\begin{equation} d \times e \mod \lambda(n) = 1 \
413 \times 17 \mod 780 = 1 \end{equation}

1. public key is (n = 3233, e = 17), with message m, encryption function is

\begin{equation} c(m) = m^{17} \mod 3233 \
c = 65^{17} \mod 3233 = 2790 \end{equation}

2. private key is (n = 3233, d = 413), with ciphertext c, decryption function is

\begin{equation} m(c) = c^{413} \mod 3233 \
m = 2790^{413} \mod 3233 = 65 \end{equation}

### Where is RSA used

• HTTPS
• SSH
• any TLS or SSL implementation
• Email
• PGP

## Key Exchanges

• RSA is all well and good
• But how do you securely exchange keys
• This is where the Diffie–Hellman key exchange comes into play

## Diffie-Hellman

• An algorithm for securely transferring keys publicly
• Simple in design, yet surprisingly robust

## Simple overview

### Generating Initial Secrets ### Derriving Shared Secrets ## Hashes

• Hashes are unpredictable, random, digests of data.
• They have a few key features:
• Random, and completely unpredictable
• One way, they cannot be easily reversed
• Transform large data into small data of a known size
• Unique, not likely that two pieces of data with the same hash

## Random

• Hashes should be sufficiently random
• They should be virtually impossible to predict

## One way

• Hashes should be impossible to reverse
• The only way to find the original data is to brute force

## Transform

• Hashes should be able to represent larger data
• They should be a known length

## Unique

• Hashes are flawed
• Converting large data to small data will result in clashes
• These are called collisions
• A good hash should reduce the likely-hood of collisions

## Uses of hashes

• What’s the use-case?
• Data/File verification
• Digital signing

## Caveats

• Not all hash algorithms are created equal
• Use different ones for your use case
• Storing passwords? Use CPU insensitive algorithms (BCrypt)
• There’s more, but that’s for another talk

## The hidden complexity

• The maths behind cryptography is fairly straightforward
• The maths is also very secure
• The implementation may not be

## Unsafe implementation

• It’s easy to implement RSA or AES yourself
• It’s easy to miss hidden complexities
• It’s hard to generate sufficiently random numbers
• It’s hard to avoid leaking information about the key

## Takeaways

• Do not implement your own crypto
• Do not implement an existing crypto
• Use an existing implementation that’s been proven to work
• Use Libsodium
• Even if you do everything right it’s pointless since the AABill passed